211 problems

Let $G$ be a connected semisimple Lie group with finite center, all of whose simple factors have real rank at least $2$. Let $\Gamma\subset G$ be a lattice, $M$ be a compact manifo…

Let $G$) be an exponential solvable Lie group, and let ${\mathcal J}_1\subseteq{\mathcal J}_2$ be closed two-sided ideals of $C^*(G)$ such that ${\mathcal J}_2/{\mathcal J}_1$ is a…

Kobayashi's conjecture. The homogeneous space

Amenability conjecture. A connected Lie group $G$ has $m(G)<\infty$ if and only if $G$ is amenable.

Lieb's generalized Wehrl conjecture. The analogue of Wehrl's conjecture should hold for at least a wide class of Lie groups.

Let $G$ be a compact Lie group, and consider the equivalence classes arising from its KAK decomposition. Fundamental-group conjecture. The classification of these equivalence class…

Margulis' conjecture. There exists a neighborhood $U\subset G(\R)$ of the identity such that for any irreducible arithmetic cocompact lattice $\Gamma\subset G(\R)$, the intersectio…

A noncompact Einstein homogeneous space is a noncompact homogeneous Riemannian manifold whose metric is Einstein, and an isometry group acts simply transitively when through every…

Matsuki–Gindikin conjecture. For every $K_\mathbb C$-orbit $S$ of nonholomorphic type on $X$, one has

Nonabelian Brunn–Minkowski conjecture. For every pair of compact sets $A,B\subseteq G$,

Let $(G,g)$ be a semi-Riemannian Lie group with a subgroup $K$ generating a left-invariant conformal foliation $\mathcal F$ on $G$ of codimension two. Semisimple subgroup minimalit…

Let $G$) be a connected, non-compact, semi-simple Lie group with finite center. Write … for the comparison map from continuous bounded cohomology to continuous cohomology. Dupont's…

Let $G$ be a Lie group with maximal torus $T$, and let $G\to G/T$ be the quotient fibration. Write $E_r^{*,*}(G;\mathbb{Z})$ for its Leray–Serre spectral sequence, and regard…

Nonpositive-curvature conjecture. Theorem 1.6 remains true under this weaker assumption: $G$ is solvable; if, in addition, $G$ is unimodular, then $\sigma=0$ and $\mathrm{g}$ is fl…

Let $G^\textbf{R}$ be a real form of a complex simple Lie group, and let $\hat{\rho}$ be a magical $\mathfrak{sl}_2$-triple of $G^\textbf{R}$. The generalized Cayley correspondence…

Monod's isomorphism conjecture. The continuous bounded cohomology of $G$ with real coefficients is naturally isomorphic to its ordinary continuous cohomology:

Let $\mathfrak{l}$ be an $n$-dimensional Lie algebra, and let $L$ be the connected and simply connected Lie group with Lie algebra $\mathfrak{l}$. The group…

Connection-blocking conjecture. The space $M$ is blockable if and only if

Non-orderability conjecture for higher-rank lattices. Lattices in simple Lie groups of rank at least two have no left invariant total order.

Let $\operatorname{Diff}(S^1)$ denote the group of diffeomorphisms of the circle $S^1$, equipped with its group structure. A Poisson-Lie structure is a Poisson structure for which…

Saturation conjecture for semisimple Lie groups. If $\lambda+\mu+\nu$ annihilates all elements of $T$ with semisimple centralizer, then

Let $G$ be a complex reductive Lie group, let $V_\lambda$ and $V_\mu$ be irreducible representations with dominant highest weights $\lambda$ and $\mu$, and let $w$ and $v$ be Weyl…

Let $G$ be a Lie group with bi-invariant metrics $h_0$ and $h_1$, let $M$ be the matrix of $h_1$ with respect to $h_0$, and let $h$ be another invariant metric. Write $\Phi$ for th…

The orthogonal abelian-subalgebra conjecture. One has $\lambda=\mu=0$. Equivalently, $\mathfrak{g}$ decomposes into three orthogonal $\Psi$-invariant abelian subalgebras. The theor…

Let $G=SO(4)$ with Lie algebra $\mathfrak{g}=so(4)=\mathfrak{g}_1\oplus\mathfrak{g}_2$, where $\mathfrak{g}_1$ and $\mathfrak{g}_2$ are the two factors. Let…